Transforming Linear Functions
2-6
LEARNING GOALS FOR LESSON 2.6
Recognize sketch and write transformations of linear functions
including (1) translation, (2) reflection, (3) stretch/compression
and (4) combination of transformations.
Translations
Horizontal
f ( x )  f ( x  h)
Vertical
f (x)  f (x)  k
Reflections
X-axis
f ( x )  f ( x )
Y-axis
f ( x )  f ( x )
Stretches/Compressions
Horizontal
1 
f (x)  f  x 
b 
Vertical
f (x)  a  f (x)
2-6
Transforming Linear Functions
Example 1: Translating & Reflecting Linear Functions
Let g(x) be the indicated transformation of f(x). Write the rule for g(x).
A.
f(x) = x – 2 , horizontal translation 3 right
f(x)
Rule:
B. f(x) = 3x + 1; horizontal translation 2 units right
f(x)
Rule:
C. f(x) = 3x + 1; vertical translation 5 units up
f(x)
Rule:
LG 2.6.1
g(x)
LG 2.6.1
g(x)
LG 2.6.1
g(x)
2-6
Transforming Linear Functions
Example 1B: Translating Reflecting Functions
Write the rule for g(x). The following
is a linear function defined in the
table; Reflect across x-axis
LG 2.6.2
x
–2
0
2
f(x)
0
1
2
Stretches and compressions change the _____ of a linear function.
If the line becomes. . .
STEEPER: stretched __________ or compressed ___________
FLATTER: compressed _________ or stretched _____________
Helpful Hint
These don’t change!
• y–intercepts in a horizontal stretch or compression
• x–intercepts in a vertical stretch or compression
2-6
Transforming Linear Functions
Example 2: Stretching & Compressing Linear Functions
LG 2.6.3
Let g(x) be a horizontal compression of f(x) = –x + 4 by a
factor of ½. Write the rule for g(x), and graph the function.
f(x)
Rule:
g(x)
Let g(x) be a vertical compression of f(x) = 3x + 2 by a factor
of ¼. Write the rule for g(x) and graph the function.
f(x)
Rule:
g(x)
2-6
Transforming Linear Functions
Ex 3: Combining Transformations of Linear Functions
LG 2.6.4
Let g(x) be a horizontal shift of f(x) = 3x left 6 units followed
by a horizontal stretch by a factor of 4. Write the rule for g(x).
Step 1 First perform the translation.
Step 2 Then perform the stretch.
f(x)
Rule:
g(x)
g(x)
Rule:
h(x)
Let g(x) be a vertical compression of f(x) = x by a factor of ½
followed by a horizontal shift 8 left units. Write the rule for g(x).
f(x)
Rule:
g(x)
g(x)
Rule:
h(x)